What is the proof behind the gradient of a curve being proportional to the distance between the two co-ordinates in the x-axis?
Don’t think about it as the $w_{(n)}-w_{(n+1)}$ being proportional to something. Think about it this way:
I'm now at $w_{(n)}$. Where do I want to be at timestep, so that the error decreases? For that, I need to know how the error changes when I make small steps to the left or right of $w_{(n)}$.
If $E$ increases as I increase $w$ (that is, if $\frac{\partial E}{\partial w}&gt;0$, then obviously, I would want to move a little bit to the left. In other words, $w_{n+1}&lt;w_{n}$ or $w_{n+1}-w_{n}&lt;0$.
On the other hand, if the derivative were negative, you know that you should move right to reduce the error a little bit, $w_{n+1}-w_{n}&gt;0$.  So, basically, your step should have the opposite sign of the derivative.
$$w_{n+1}-w_{n} \propto-\frac{\partial E}{\partial w}$$
$\alpha$, the learning rate, is just the constant of proportionality.
Caution: think about small values for this rate, not big numbers.  Taking a huge step can cause you to overshoot the minimum point.